MathDB

5

Part of 2022 South East Mathematical Olympiad

Problems(2)

find the minimum

Source: 2022 China Southeast Grade 10 P5

8/3/2022
Positive sequences {an},{bn}\{a_n\},\{b_n\} satisfy:a1=b1=1,bn=anbn−1−14(n≥2)a_1=b_1=1,b_n=a_nb_{n-1}-\frac{1}{4}(n\geq 2). Find the minimum value of 4b1b2⋯bm+∑k=1m1a1a2⋯ak4\sqrt{b_1b_2\cdots b_m}+\sum_{k=1}^m\frac{1}{a_1a_2\cdots a_k},where mm is a given positive integer.
inequalitiesNumber sequenceGCD
China South East Mathematical Olympiad 2022 Grade11 Q5

Source: Jianxi jian

8/3/2022
Let a,b,c,da,b,c,d be non-negative integers. (1)(1) If a2+b2−cd2=2022,a^2+b^2-cd^2=2022 , find the minimum of a+b+c+d;a+b+c+d; (1)(1) If a2−b2+cd2=2022,a^2-b^2+cd^2=2022 , find the minimum of a+b+c+d.a+b+c+d .
inequalitiesalgebranumber theory